91Ó°ÊÓ

Find \(y^{\prime}\). $$\text { If } y=\sqrt[3]{x}+\sqrt{x}, \text { find }\left.\frac{d y}{d x}\right|_{x=64}$$

Find \(\frac{d y}{d x}\) for each pair of functions.\( Find \)\frac{d y}{d t}\( if \)y=\frac{1}{3 u^{5}-7}\( and \)u=7 t^{2}+1$

Is the function given by \(f(x)=\left\\{\begin{array}{ll}\frac{1}{3} x+4 & \text { for } x<3, \\ 2 x-1 & \text { for } x \geq 3.\end{array}\right.\) continuous at \(x=3 ?\) Why or why not?

Find \(y^{\prime}\). $$\text { If } y=\frac{1}{3 x^{4}}, \text { find }\left.\frac{d y}{d x}\right|_{x=-1}$$

The function \(p(t)=\frac{2000 t}{4 t+75}\) models the population \(p\) of deer in an area after \(t\) months. a) Find \(p^{\prime}(10), p^{\prime}(50),\) and \(p^{\prime}(100)\) b) Find \(p^{\prime \prime}(10), p^{\prime \prime}(50),\) and \(p^{\prime \prime}(100)\) c) Interpret the meaning of your answers to parts (a) and (b). What is happening to this population of deer in the long term?

Find \(\frac{d y}{d x}\) for each pair of functions.\( Find an equation for the tangent line to the graph of \)y=\sqrt{x^{2}+3 x}$ at the point (1,2).

Find \(\frac{d y}{d x}\) for each pair of functions.\( Find an equation for the tangent line to the graph of \)y=\left(x^{3}-4 x\right)^{10}$ at the point (2,0).

A medication is injected into the bloodstream, where it is quickly metabolized. The percent concentration \(p\) of the medication after \(t\) minutes in the bloodstream is modeled by the function \(p(t)=\frac{2.5 t}{t^{2}+1}\) a) Find \(p^{\prime}(0.5), p^{\prime}(1), p^{\prime}(5),\) and \(p^{\prime}(30)\) b) Find \(p^{\prime \prime}(0.5), p^{\prime \prime}(1), p^{\prime \prime}(5),\) and \(p^{\prime \prime}(30)\) c) Interpret the meaning of your answers to parts (a) and (b). What is happening to the concentration of medication in the bloodstream in the long term?

Find \(\frac{d y}{d x}\) for each pair of functions.\( Find an equation for the tangent line to the graph of \)y=x \sqrt{2 x+3}$ at the point (3,9).

Use a graphing calculator to check the results of Exercises \(1-48\).